We consider a multiple failure time study containing \(n\)  independent subjects. For the \(i\textsuperscript {th}\)  subject, there exist three sets of \(d_k\) -dimensional vectors of covariates \(Z_{ik}=(Z_{i1},\ldots ,Z_{id_k})^\top \)  for \(k=1,2,3\)  and \(i=1,\ldots ,n\) . \(T_{i1}\)  and \(T_{i2}\)  denote the time to the non-terminal and terminal events, respectively, for the \(i\textsuperscript {th}\)  subject.
One key concept that needs to be characterized formally in illness-death models is realizing the form of dependence between \(T_1\)  and \(T_2\) . A popular procedure for structuring this dependence is introducing a frailty term [1]}, [2]}, [3]} to absorb the information on the dependence of non-terminal and terminal events.
